# Yulia's Dream Research Papers

Yulia's Dream is a free math enrichment and research program for exceptional high school students (grades 10-11) from Ukraine.

## Research Papers

### 7) Ivan Bortnovskyi, Vasily A. Dolgushev (Temple University), Borys Holikov, Vadym Pashkovskyi, First examples of non-abelian quotients of the Grothendieck-Teichmueller group that receive surjective homomorphisms from the absolute Galois group of rational numbers (arXiv.org, 20 May 2024)

Many challenging questions about the Grothendieck-Teichmueller group, $GT$, are motivated by the fact that this group receives the injective homomorphism (called the Ihara embedding) from the absolute Galois group, $G_Q$, of rational numbers. Although the question about the surjectivity of the Ihara embedding is a very challenging problem, in this paper, we construct a family of finite non-abelian quotients of $GT$ that receive surjective homomorphisms from $G_Q$. We also assemble these finite quotients into an infinite (non-abelian) profinite quotient of $GT$. We prove that the natural homomorphism from $G_Q$ to the resulting profinite group is also surjective. We give an explicit description of this profinite group. To achieve these goals, we used the groupoid $GTSh$ of $GT$-shadows for the gentle version of the Grothendieck-Teichmueller group. This groupoid was introduced in the recent paper by the second author and J. Guynee and the set $Ob(GTSh)$ of objects of $GTSh$ is a poset of certain finite index normal subgroups of the Artin braid group on 3 strands. We introduce a sub-poset $Dih$ of $Ob(GTSh)$ related to the family of dihedral groups and call it the dihedral poset. We show that each element $K$ of $Dih$ is the only object of its connected component in $GTSh$. Using the surjectivity of the cyclotomic character, we prove that, if the order of the dihedral group corresponding to $K$ is a power of 2, then the natural homomorphism from $G_Q$ to the finite group $GTSh(K, K)$ is surjective. We introduce the Lochak-Schneps conditions on morphisms of $GTSh$ and prove that each morphism of $GTSh$ with the target $K$ in $Dih$ satisfies the Lochak-Schneps conditions. Finally, we conjecture that the natural homomorphism from $G_Q$ to the finite group $GTSh(K, K)$ is surjective for every object $K$ of the dihedral poset.

### 6) Darij Grinberg (Drexel University), Nazar Korniichuk, Kostiantyn Molokanov, and Severyn Khomych, The Pak–Postnikov and Naruse skew hook length formulas: A new proof (2 Oct 2023; arXiv.org, 27 Oct 2023)

The classical hook length formula of enumerative combinatorics expresses the number of standard Young tableaux of a given partition shape as a single fraction. In recent years, two generalizations of this formula have emerged: one by Pak and Postnikov, replacing the number by a (rational) generating function, and one by Naruse, which generalizes the setting from a partition to a skew partition. Both generalizations appear to lie significantly deeper, with no simple proofs known. We combine them into a generating-function identity for skew partitions, and prove it in a fairly elementary way using recursion, determinants and simple combinatorics.

### 5) Dmytro Antonovych, Viktor Makozyuk, Vladyslav Tysiachnyi, Matching of frames of open Jacobi diagrams and chord diagrams (31 Jul 2023)

This paper is dedicated to the study of similarities between two objects that arise from the theory of Vassiliev invariants: open Jacobi diagrams and chord diagrams, which are uni-trivalent graphs with "orientation" structure and trivalent graphs with the structure of the chosen Hamilton cycle, respectively. We define a space of open Jacobi diagrams as the span of Jacobi diagrams modulo IHX and AS relations and a space of chord diagrams as a span of chord diagrams modulo 4T relations. We define two "frame" maps as operations from sets of chord and Jacobi diagrams to trivalent graphs. These operations act by forgetting about the structures in Jacobi and chord diagrams, making them trivalent graphs. Using these operations, we find the correspondence between the frames of elements of bases of spaces of Jacobi diagrams and chord diagrams, and formulate the "Frame matching" conjecture. As an intermediate step for proving the "Frame matching" conjecture, we prove a useful lemma, which states that any open Jacobi diagrams can be presented as a linear combination of special open Jacobi diagrams, which we called chord type Jacobi diagram. Finally, we wrote Python code to check the "Frame matching" conjecture; we provide an algorithm of how to use this code for checking the "Frame matching" conjecture.

### 4) Semen Andriets, Borys Holikov, Partial orderings of minors in the positive Grassmannian (24 Jul 2023)

In this paper, we research the partial order of minors of the positive Grassmannian (i.e. a space that parametrizes $k \times n$ matrices with all $k \times k$ minors positive) with a fixed maximal set of largest minors. This is connected to a combinatorial structure called a circuit graph. The maximal set of the largest minors corresponds to some cycle of length $n$ in the circuit graph, and then, abstractly, the further away in the graph some minor is from maximal the less its value. But this further is some non-trivial property called cubical distance, which be explained in the paper.

### 3) Alexander Borodin, Martin Leshko, Marina Spektrova, Combinatorial Hikita Conjecture (7 Jul 2023)

The Hikita conjecture is connected with geometry and representation theory. There are some generalisations, but all of them conjecture some isomorphism between algebras. Combinatorial objects we investigate, in some sense, enumerate their bases, and our general task, loosely speaking, is to find \textit{combinatorial} bijections. In particular, in this article, we will formulate and prove parabolic conjecture in combinatorial form. The advantage of this method over the previously known ones is the simplicity of the used objects, which means that this fact is accessible to a larger number of readers as well as it helps to understand the algebraic theorems better.

### 2) Ivan Bortnovskyi, Vadym Pashkovskyi, Exploration of the Grothendieck-Teichmueller ($GT$) shadows for the dihedral poset (4 Jul 2023)

Grothendieck-Teichmueller ($GT$) shadows are morphisms of the groupoid $GTSh$ and they may be thought of as approximations of elements of (the gentle version of) the Grothendieck-Teichmueller group $GTh$. The set $Ob(GTSh)$ of objects of $GTSh$ is the poset of certain finite index normal subgroups of the Artin braid group on 3 strands. In this note, we introduce a subposet $Dih$ of $Ob(GTSh)$, call it the dihedral poset, and investigate connected components of the groupoid $GTSh$ for elements of this poset. We prove that every $K \in Dih$ is the only object of its connected component $GTSh_{conn}(K)$ in the groupoid $GTSh$ (in particular, $GTSh_{conn}(K)$ is a finite group). We describe the set of morphisms of $GTSh_{conn}(K)$ explicitly and we show that, for every pair $N, K \in Dih$ such that $K \le N$, the natural map $GTSh_{conn}(K) \to GTSh_{conn}(N)$ is surjective.

### 1) Yehor Avdieiev and Alexander Tsymbaliuk (Purdue University), Affine Standard Lyndon words: A-type (arXiv.org, 25 May 2023)

We generalize an algorithm of Leclerc describing explicitly the bijection of Lalonde-Ram from finite to affine Lie algebras. In type $A_n^{(1)}$, we compute all affine standard Lyndon words for any order of the simple roots, and establish some properties of the induced orders on the positive affine roots.

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